Synchronisation of almost all trajectories of a random dynamical system

TL;DR

This paper investigates conditions under which a random attracting set in a stochastic dynamical system reduces to a single point, extending previous results to more general spaces without compactness assumptions.

Contribution

It provides necessary and sufficient conditions for the attracting set to be a singleton in non-compact Lusin metric spaces, broadening the scope of earlier results.

Findings

Conditions for singleton attractors are established.

Results apply to general Lusin metric spaces, not just compact ones.

The paper extends the theory of random attractors in stochastic systems.

Abstract

It has been shown by Le Jan that, given a memoryless-noise random dynamical system together with an ergodic distribution for the associated Markov transition probabilities, if the support of the ergodic distribution admits locally asymptotically stable trajectories, then there is a random attracting set consisting of finitely many points, whose basin of forward-time attraction includes a random full measure open set. In this paper, we present necessary and sufficient conditions for this attracting set to be a singleton; our result does not require the state space to be compact, but holds on general Lusin metric spaces.